Relative
Permeability
of
Petroleum
Reservoirs
Authors
Mehdi
Honarpour
Associate
Professor
of Petroleum
Engineering
Department of
Petroleum
Engineering
Montana College
of Mineral Science
and
Technology
Butte,
Montana
Leonard
Koederitz
Professor
of Petroleum
Engineering
Department
of
Petroleum Engineering
University of
Missouri

where
q
represents
the rate at
which water
flows downward
through a
vertical sand
pack
with cross-sectional
area
A
and
length
L; the terms
h,
and
h, represent
hydrostatic
heads at
the
inlet and outlet,
respectively,
of the sand
filter, and
K is a constant.
Darcy's experiments
were confined to
the flow
of water through

is a
property
of the
fluid
(viscosity).
With this
modification,
Darcy's
law may be
written in
a
more
general
form
AS
k
l-
dz
dPl
u':*LPgos-dsl
where
S
v
Distance
in direction
of flow,
which is taken as
positive
Volume of
flux across

q/A, where
q
is the volumetric
flow
rate
and A
is the average
cross-sectional
area
perpendicular to the
lines of flow.
It can
be shown
that the
permeability term
which appears
in Darcy's
law has units
of
length squared.
A
porous
material
has a
permeability of
I D when a single-phase
fluid with
a
viscosity of
I cP completely

medium.
Since this
condition
is
seldom
met
in
a
hydrocarbon
reservoir,
it is evident
that
further
modification
of Darcy's
law is needed
if the
law is to be
applied to
the flow
of fluids
in
an
oil or
gas
reservoir.
A
more useful
form of
Darcy's law can

to the absolute
permeability of the
rock.
Effective
permeability to each
fluid
phase
is considered
to be
independent of the
other
fluid
phases
and the
phases
are
considered
to
be immiscible.
If
we
define
relative
permeability as the
ratio of
effective
permeability to absolute
perme-
ability,
Darcy's

I
Vo.:T(0.,*K-*)
V*.:*(o-'13-t)
Vo,:H(o-r#-k)
Dr. lfcL
lhc
\ltntrna
.{r(arrnl
hrr
r\rfi.Rr{le
I
tnLlt.rs
t>
nl
rstn :
rrrluhng
drc
h
t-;xrlrr
Ti
lrrya
I
\lrsr.n.R.i
R.{1.
[}r }ri
(-}rrrrrrr.n
r I
rcrtr
rrltcrj
t

permeabilities
to
the
three
fluid
phases
at
the
respective
saturations
of the
phases
within
the
rock'
Darcy's
law
is the
basis
for
almost
all
calculations
of
fluid
flow
within
a
hydrocarbon
reservoir.

the
range
of
fluid
saturations
that
will be
encountered.
The
problems
involved
in
measuring
and
predicting
relative
permeability
have
been
studied
by
many
investigators.
A
summary
of
the
major
results
of

:.:
tn
lllt'a\uring
[r ::
-:
'\
ruilflrof)'
Plc:.
THE AUTHORS
Dr. Mehdi
"Matt"
Honarpour
is
an
associate
professor
of
petroleum
engineering at
the
Montana College
of Mineral Science
and
Technology,
Butte, Montana.
Dr. Honarpour
obtained
his B.S., M.S., and
Ph.D.
in

Society of
Petroleum
Engineers of
AIME, the
honorary society of Sigma
Xi,
Pi
Epsilon Tau and
Phi Kappa
Phi.
Leonard
F. Koederitz
is a
Professor
of Petroleum
Engineering at
the University
of
Missouri-Rolla.
HereceivedB.S.,
M.S., andPh.D.
degrees
fromtheUniversityof
Missouri-
Rolla.
Dr. Koederitz
has worked
for Atlantic-Richfield
and
previously served as Department

Gas
Council and the
Petroleum Engineering
Department at the University of
Missouri-Rolla.
ACKNOWLEDGMENT
The
authors wish
to acknowledge
the Society of Petroleum
Engineers and
the American
Petroleum
Institute
for granting
permission
to use their
publications.
Special thanks are due
J. Joseph
of Flopetrol
Johnston
and
A.
Manjnath of Reservoir Inc.
for their
contributions
and
reviews
throughout

n
I
r|
n.j thc
Anrerican
li
:::.,nk.
are
due
rr:
-
'ntributions
TABLE
OF CONTENTS
Chapter
I
Measurement
of
Rock
Relative
Permeability
.
I.
Introduction.
. .
il.
Steady-State
Methods

.

6
8
9
10
t2
III.
IV.
V.
VI.
Unsteady-
State
Methods
Capillary
Pressure
Methods
Centrifuge
Methods
Calculation
from
Field Data
.
References.

Chapter
2
Two-Phase
Relative
Permeability

15

. .20
VII.
Wahl, Torcaso,
and
Wyllie
VIII.
Brooks and
Corey
. . . .27
XIIX.
Wyllie, Gardner,
and
Torcaso
. . .
.
.29
X.
Land,
Wyllie,
Rose,
Pirson,
and
Boatman

30
XI.
Knopp,
Honarpour
et al.,
and

50
V. Definition
and Causes
of
Wettability.
54
VI.
DeterminationofWettability
58
A. Contact
Angle Method
58
B.
ImbibitionMethod.
60
C.
Bureau of
Mines
Method
63
D. Capillarimetric
Method
63
E.
FractionalSurfaceAreaMethod
64
F.
Dye
Adsorption
Method

66
M.
Relative Permeability
Method

66
N.
Relative
Permeability
Summation
Method
61
O.
Relative
Permeability
Ratio
Method
67
P.
Waterflood
Method
68
a.
Capillary
Pressure
Method

.
68
R.

of Porosity
and
Permeability
79
XII.
Effects
of Temperature.
. .82
XIII.
Effects
of Interfacial
Tension and
Density
. . .82
XIV. Effects
of
Viscosity
.
.;
. ' ' 83
XV. Effects
of
Initial
Wetting-Phase
Saturation
89
XVI.
Effects
of an
Immobile

and
Wyllie
105
C.
Reid.

107
D.
Snell.

l0g
E.
Donaldson
and
Dean

. . I l0
F.
Sarem
113
G.
Saraf
and
Fatt
I 15
H.
WyllieandGardner
.'ll5
m.
Imbibition

Tests

' 132
VIII. ComparisonofModels
'133
References""'
"""'134
Appendix
Symbols.

137
Tbc
I
hr crth
r3th\
rrl
c{ehlr.
\,ilUt-3ll
irlurltl
thc crr
Itrf
ft\
thc
Ha
ln
tt
thc
tc.
drrqlg
urcfrr|

flt
Itr
lfi'
rnarl
ln ci
r-all,
thYl.
6-i
66
66
6-
6-
6,\
hs
h\
6\
-:
l
Chapter
I
MEASUREMENT OF
ROCK RELATIVE PERMEABILITY
I.
INTRODUCTION
The
relative
peffneability
of a
rock
to each

effects
at the outflow boundary
of
the core. Steady-state methods are
preferred
to unsteady-state methods by some investigators
for
rocks of intermediate
wettability,'
although some difficulty
has
been reported in applying
the
Hassler
steady-state method to this type
of rock.2
ln
the capillary
pressure
method, only the nonwetting
phase
is injected into
the core during
the test. This fluid displaces the
wetting
phase
and the
saturations
of both
fluids

et a1.,7 and Geffen
et al.8 The
version of the apparatus
which was described by Geffen
et al., is illustrated by
Figure
l. In
order
to reduce end effects
due to capillary
forces, the sample to be tested is
mounted between two
rock samples which
are similar to the test
sample. This
arrangement
also
promotes
thorough
mixing of the
two fluid
phases
before they enter the test sample.
The laboratory
procedure is
begun
by saturating the
sample with one fluid
phase
(such

is reached, the two flow
rates
are
recorded and the
percentage
saturation of each
phase
within the test sample
is
determined by removing the test sample
from the assernbly and
weighing it. This
procedure
introduces
a
possible
source of experimental error,
since a small amount
of fluid may be
lost because of
gas
expansion and
evaporation. One authority
recommends that the core be
wgighed under oil, eliminating
the
problem
of obtaining the
same
amount

which have been used for in situ determination
of fluid saturation
in cores include
measurement
of electric
capacitance, nuclear
magnetic resonance, neutron
scattering,
X-ray
absorption,
gamma-ray
absorption,
volumetric
balance,
vacuum distilla-
tion, and microwave techniques.
.le
Relative Permeabilin of
Petroleum
Reservoirs
El-ectrodes
Outl-et
Differential
Pressure
Taps
Inlet
Inlet
FIGURE
l. Three-section core assembly.8
After fluid

permeability
at either increasing
or decreasing saturations
of the wetting
phase
and it can be applied
to both
liquid-liquid
and
gas-liquid
systems. The direction
of
saturation
change used
in
the laboratory should cor-
respond to field conditions.
Good capillary contact between the test sample
and the adjacent
downstream core is
essential
for
accurate
measurements
and temperature must be held
constant during the test. The
time
required for
a test to
reach

by
Figure
1), the two fluid
phases
are
injected
simultaneously through a
single
core. End effects are minimized
by using
relatively high flow rates,
so the region of high
wetting-phase
saturation at the outlet face of the core is small. The theory which was
presented
by Richardson et al. for describing
the
saturation distribution within
the core
may
be de-
veloped
as
follows. From Darcy's law, the
flow of two
phases
through a horizontal linear
system can be
described by the equations
-dP*,

where the subscripts wt
and
n
denote the
wetting
and
nonwetting
phases,
respectively. From
the definition of capillary
pressure,
P", it follows
that
1.0
o
a
0
lel

.
ICsr-
J
ii-
*i'trDd
CE'.i-:;
ir
[C-
plcir
:Jtrtr\r'
3T

BJ ,,.:l'l!ls'
f3h
"
: nrsh
Jil.
l-:
s'ntcrj
!
n-:.
re'
Jc-
iz '.
a(rr
5 10
15
20
25
Distance
from Outflow
Face,
cffi
FIGURE 2.
Comparison
of saturation
gradients
at low
flow rate.e
dP.:dP dP*,
These three equations
may be

(s)
(6)
dS*,
dL
|
/Q*,
Fr*,
Q"p.\
I
:A\
k*
-
L"
/op.rus*
,lt
Richardson et
al. concluded
from experimental
evidence
that the nonwetting
phase
sat-
uration at
the discharge
end of
the core
was at
the equilibrium
value,
(i.e.,

is
virtually constant
from the
inlet
face to a
region a
few centimeters
from the
outlet.
Within
this
region the
wetting
phase saturation
increases to the equilibrium
value at the
outlet
face.
Both
calculations
and experimental
evidence
show that
the region
of high
wetting-phase
saturation
at
the discharge
end

saturation
gradient
f nf low f ace
1>
Relative
Permeability
of
Petroleum
Reservoirs
1.0
\o't
I
-o-o-
-o o-
- :-

:
-
J
t
Theoretical
saturation
gradient
Inf row
rac"
a>l
o
5
10
15

discharge
end
of the
core,
excessive
rates
must
be
avoided.
Problems
which
can
occur
at
very
high
rates
include
nonlaminar
flow.
C.
Stationary
Fluid
Methods
Leas
et al.12
described
a technique
for
measuring

was
modified
slightly
by
Osoba
et
al.,s
who
held
the
liquid phase
stationary
within
the
core
by
means
of
barriers
which
were
permeable
to
gas
but not
to the
liquid.
Rapoport
and
Leasr3

using
barriers
which
were
permeable
to water
but impermeable
to oil
and gas.
Osoba
et
al.
observed
that
relative permeability
to
gas
determined
by
the
stationary
liquid
method
was
in
good
agreement
with
values
measured

methods
in
the
region
of
equilibrium
gas
saturation.
This
situation
resulted
in
an
equilibrium
gas
saturation
value
which
was
higher
than
obtained
by
the
other
methods
used
(Penn-Siate,
Single-Sample
Dynamic,

for
relative permeability
measurement
which
was
described
by
Hasslerrs
in 1944.
The
technique
was
later
studied
and
modified
by
Gates
and
Lietz,16
Brownscombe
et
?1.,"
Osoba
et
al.,s
and
Josendal
et
al.ro

of
the
core,
but
allow
both
phases
to
flow
simultaneously
through
the
core.
The pressure
o
a
ns:ii
tu
t'br
cr';rd
crl
n
.trlf!
fl:e
rc
;rrbt
lrl
tw.l.
Ilr l{rr
4rS

i,'.j.
li
.
-,
'
\v\3
Jl
*'

ACrC
l&'
- .
-:.rquc
'h-
"
- :q;rJ
ft .ii'n'l
3a.
.
.h:
Cl
lqL.
:
-,crlxrJ
I
o:
:-i

J
ti^ t
l{e ler
lct
-,-:
'xrtlc't
Th.
:-i; urc'
FIGURE
4.
Two-phase relative
permeability apparatus.r5
in each
fluid
phase
is measured
separately through
a semipermeable
barrier.
By
adjusting
the flow
rate of the
nonwetting
phase,
the
pressure
gradients
in the
two

wet
by one
of the fluids, but
some difficulty
has been
reported
in using the
procedure under conditions
of
intermediate
wettability.2'r8
The
Hassler
method
is not widely used
at this
time, since the
data can be obtained
more
rapidly
with other
laboratory
techniques.
E.
Hafford
Method
This steady-state
technique
was described
by Richardson

of the disc
is used to measure
the
pressure
in the
wetting
fluid at the
inlet of the
sample. The
nonwetting
fluid is injected directly
into the sample and
its
pressure
is measured through
a standard
pressure
tap
machined into the
Lucite@ sur-
rounding the sample.
The
pressure
difference between
the
wetting and the nonwetting
fluid
is a
measure of the
capillary

the Hafford and
single-sample dynamic
meth-
Relative
Permeabilin
of Petroleum Reservoirs
GAS
I
GAS
PRESSURE
GAUGE
PRESSURE
GAS
METER
OIL BURETTE
FIGURE
5.
Hafford relative permeability
apparatus.e
ods.
In
the dispersed
feed
method,
the wetting
fluid
enters
the test sample
by first passing
through

that
it
enters
the test sample
more
or less
uniformly
over the inlet
face.
The
nonwetting
phase
is
introduced
into radial grooves
which
are machined
into
the
outlet face
of the
dispersing
section,
at the
junction
between
the
dispersing material
and
the test sample.

is
more
difficult. The
theory
developed
by Buckley and Leverettre
and extended
by
Welge2o
is
generally
used for
the measurement
of
relative permeability
under
unsteady-state
conditions.
The
mathematical
basis for interpretation
of the
test data
may be
summarized
as follows:
Leverett2r
combined
Darcy's
law

the angle
between
direction x
and horizontal;
and
Ap is
the density
PRESSURE
rtl
.r[I
t
'.lt
Sn
I
.
t
!|t
llE
3
^
-
G€
I
t
-:.'iJr
-
il.1::
-
:a\re
f

llrr<'
bt*
-,-:l
:l
lrr-
''
j\'
:\
le
.:
.:t":.to!
Er
-::,i.cfilr
J
:
-, :c
r\
.(#)
/,(a
7
difference between displacing and displaced
fluids. For
the case of horizontal flow
and
negligible capillary
pressure,
Welge2o
showed that
Equation
7

the
slope of a
plot
of
Q*
as
a function of S*,ou.
By
definition
l,z:q,,/(q,,*q*)
By combining this
equation with Darcy's
law, it can be shown that
I
f,,r:
'
tlOt
I1.,/
K ,
t
*
tr/.,*
Since
p"
and
pw
are known, the relative
permeability
ratio k.o/k.*
can be determined from

t.z
ttr.
where I,,
the ?elative injectivity, is
defined as
(
I l)
(12)
(
l3)
I,:
injectivity
initial
injectivity
(q*,/Ap)
(q*,/Ap)
at start of
injection
A
graphical
technique
for solving Equations 1l and 12 is illustrated in Reference L3
Relationships describing relative
permeabilities
in a
gas-oil
system may be obtained
by
replacing
the subscript

The core be homogeneous.
The driving force and fluid
properties
be held constant during
the test.2
l.
2.
3.
4.
Relative
Permeabilin of
Petroleum
Reservoirs
Laboratory
equipment
is
available for making the unsteady-state
measurements
under sim-
ulated
reservoir
conditions.2a
In
addition to the JBN method, several alternative techniques for determining relative
permeability
from
unsteady-state test data
have
been
proposed.

method for
determining two-phase
relative
permeability
and capillary
pressure
from
two
sets of displacement
experiments,
one set conducted at a
high
flow rate and the other at a
rate representative
of reservoir conditions.
The
theory
of Welge was
extended by Sarem to
describe relative
permeabilities
in a system containing three fluid
phases.
Sarem employed
a
simplifying
assumption
that the
relative
permeability

or
natural water
drive;
kr/k"
is employed to
estimate the
production
which will be
obtained
from recovery
processes
where
oil is displaced
by
gas,
such as
gas
injection or solution
gas
drive. An important use of
the
ratio k*/k*
is in the
prediction
of
performance
of natural
gas
storage
wells,

permeability
rock. For these types of
reservoirs it may
be advisable
to measure
k*/k., in
cores
which
contain an
immobile water
saturation.2a
IV. CAPILLARY PRESSURE METHODS
The
techniques which are
used
for
calculating
relative
permeability
from capillary
pressure
data were
developed
for
drainage situations,
where a
nonwetting phase
(gas)
displaces a
wetting phase

in a water-
wet rock
is an imbibition
process
rather than a drainage
process.
Although
capillary
pressure
techniques
are
not usually the
preferred
methods for
generating
relative
permeability
data,
the
methods
are useful
for
obtaining
gas-oil
or
gas-water
relative
permeabilities
when rock samples
are too

estimating
kr/k"
ratios for
retrograde
gas
condensate reservoirs,
where oil
saturation increases
as
pressure
decreases,
with
an
initial
oil saturation which may be as low as zero. The capillary
pressure
methods
are recommended
for this situation because the conventional unsteady-state
test
is not
de-
signed for very
low oil saturations.
Data obtained
by
mercury injection are customarily
used when relative
permeability
is

-'.
Jc-
flc"
-

.:ilCr-
J
i* :cicJ
to
d
R
.zcllc':'
3fi:.;:^illlts'r
rl
:rhcd
fE
" 'l'.
l\Atr
lr
"'-':
r[ r
f'.
:.::;::l
ltr
!n
"
':'
",cJ
J
r

-, llr.rl
l.
,:.it.
r
-'r-\,r'
D'-
.*
-:lJl
br.
:':; urc
)
;
.,
.: '\
J
bJ'
: rrll
.1
"
'-
rr
br-,
'-
i-,rc
f
i-
^
:iS[-
!l
:;

lr;h
is
9
the
nonwetting
phase)
is injected
in
measured increments
at
increasing
pressures. Approx-
imately
20 data
points
are
obtained
in a typical
laboratory test
designed to
yield
the complete
capillary
pressure curve,
which
is required
for calculating
relative
permeability
by the meth-

(
l4)
and
(
l5)
where
the
subscripts
wt
and
nwt denote the
wetting and
nonwetting
phases,
respectively,
and
n has a
value
of
2.0. Fatt and
Dykstra3o developed
similar equations
with
n
equal
to
3.0.
A slightly different
result
is

duced
from
rock samples
which were
initially saturated
uniformly
with one
or two
phases.
Liquids are collected
in transparent
tubes connected
to the rock
sample holders and
production
is monitored throughout
the test.
Mathematical
techniques
for deriving
relative
permeability
data
from these
measurements
are described
in References
26, 27, and
28.
Although

subject to
capillary
end effect
problems
and they
do not
provide
a
means for
determining
relative
permeability to the
invading
phase.
O'Mera
and
Lease28 describe
an automated
centrifuge
which employs a
photodiode
array
in
conjunction
with a
microcomputer
to
image and
identify
liquids

are square
in cross
section,
since
a cylindrical
tube would
act as
a
lens
and concentrate
the light
in a narrow
band
along
the major
axis of
the tube. A
schematic
diagram of
the apparatus
is shown
by Figure
6.
VI.
CALCULATION
FROM FIELD
DATA
It is
possible
to calculate

(free
gas)
*
(solution
gas)
(18)
If
we
consider
the
flow
of free
gas
in
the reservoir, Darcy's
law
for
a radial
system may
be
written
SrmrLrlr
Thll. tu
rtts:rt
r.
\&trt
or
fra
g,
;rrrrrrhrr

-w
FrB,
ln
(r./r*)
(
l9)
COMPUTER
o
z
LIJ
o
o
uJ
LIJ
o-
a)
o
U'
IJJ
tr
o
o
J
:
CONTROLLER
SPEED
SET
POINT
ll
?

gas
formation
volume
factors, respectively.
The ratio
of free
gas
to oil
is obtained
by
dividing
Equation
19 by
Equation
20. lt
we
express
Ro,
cumulative
gas/oil
ratio and
R,, solution
gasioil ratio, in terms
of standard
cubic
foot
per
stock tank
barrel,
Equation

F.
l|i
'':J:1Jac\
!n
\e\ll()n.
I
^-:lJ
rltr0S
I
F
,:l
6
l:
1' :::rltng
Dd
:l
the
l\r
|
:',
.t::t
tx?)
The oil
saturation
which corresponds
to this
relative
permeability ratio may be determined
from a material
balance.

N denotes
initial
stock tank barrels
of oil
in
place;
No is number
of
stock tank
barrels of oil
produced;
and B",
is the ratio of the
oil volume at
initial reservoir
conditions
to oil
volume at standard
conditions.
If total
liquid saturation
in the
reservoir
is expressed as
(23)
s,:s*+(r-s*)(\})
(*)
(24)
rl9t
then the

is useful even
if only a
few high-
liquid-saturation
data
points
can be
plotted.
These
kr/k" values can be used to
verify the
accuracy of
relative
permeability
predicted
by empirical
or laboratory
techniques.
Poor
agreement between
relative
permeability determined
from
production
data and
from
laboratory
experiments
is not uncommon.
The causes

pressure
and saturation
gradients
which
are
present
in
the reservoir,
nor does
it allow for
the fact that wells may
be
producing
from
several strata which
are at various
stages of depletion.
3. The
usual
technique for calculating relative
permeability
from field
data assumes
that
Ro
at any
pressure
is constant
throughout the oil
zone.

REFERENCES
l.
Gorinik, B. and Roebuck,
J.
F.,
Formation Evaluation
through
Extensive
Use of
Core Analysis,
Core
Laboratories,
Inc.,
Dallas, Tex.,
1979.
2.
Saraf, D. N.
and McCaffery,
F.
G.,
Two-
and
Three-Phase
Relative
Permeabilities:
a
Review,
Petroleum
Recovery
Institute Report

A.,
and
Blair,
P. M., Laboratory
relative
permeability
measurements,
Trans.
AIME, 192, 47, 1951.
6.
Henderson,
J.
H.
and Yuster, S.T.,
Relative
permeability
study,World
Oil,3,139, 1948.
7. Caudle,
B. H.,
Slobod, R. L.,
and Brownscombe, E.
R. W., Further
developments in
the laboratory
determination
of relative
permeability,
Trans. AIME,
192, 145,

permeability,
Trans.
AIME, 195,
187, 1952.
10.
Josendal,
V. A.,
Sandiford, B.
B., and Wilson,
J.
W., Improved multiphase
flow
studies employing
radioactive
tracers,
Trans. AIME,
195, 65, 1952.
I l. Loomis,
A.
G. and
Crowell,
D.
C., Relative Permeability
Studies:
Gas-Oil and Water-Oil
Systems,
U.S.
Bureau
of Mines Bulletin
BarHeuillr,

H., and Wyllie,
M. R.
J.,
Three-phase relative perme-
ability,
J.
Pet.
Technol., Nov.,
63, 1956.
15.
Hassler,
G. L., U.S. Patent
2,345,935,
1944.
16.
Gates,
J.
I. and Leitz,
W. T., Relative permeabilities
of
California cores
by the capillary-pressure
method,
Drilling
and Production
Practices,
American Petroleum
Institute, Washington,
D.C. 1950,
285.

of fluid displacement
in sands,
Trans. AIME,
146,107,
1942.
20.
Welge'H.J.rAsimplifiedmethodforcomputingrecoverybygasorwaterdrive,Trans.A|ME,
195,91,
1952.
21. Leverett,
M.
C., Capillary
behavior in
porous
solids,
Trans. AIME,
142, 152, 1941.
tl
l_1
lo
ll
:.i
Ir
Johr
plar'cn
Crid
Clrfi
SFr t.
Jcrl
.lr.plr

lr
[1gI
tE ::tC.
thal
D
:i' .c.rJ
to
lll
-: l
I|r.
.
-:.c tli
Fn
:-'-<'I\t)lf.
I
c\:r J
lrr)m
rl
lE-
F.
lr-
X'r
|
:',
F
I.
Er
ls
It
!

Prentice-Hall,
Englewood
Cliffs,
1977,
chap. 7.
24.
Special Core Analysis,
Core Laboratories,
Inc., Dallas,
1976.
25.
Jones,
S.
C. and Roszelle, W.
O., Graphical
techniques for
determining
relative permeability
from
displacement experiments,
J.
Pet.
Technol.,
5, 807, 1978.
26.
Slobod, R. L.,
Chambers, A.,
and Prehn, W. L.,
Use of
centrifuge for

and Lease, W.
O.,
Multiphase
relative
permeability
measurements
using an automated
centrifuge,
Paper
SPE
12128 presented
at the SPE 58th
Annual Technical
Conference
and Exhibition,
San
Francisco.1983.
29. Purcell,
W. R.,
Capillary
pressures
-
their measurement
using mercury
and the
calculation
of
permeability
therefrom,
Trans. AIME,

RELATIVE
PERMEABILITY
I. INTRODUCTION
Direct
experimental
measurement
to determine
relative
permeability of
porous rock has
long
been
recorded
in
petroleum related
literature.
However,
empirical
methods for deter-
mining
relative
permeability
are
becoming
more
widely used,
particularly with the
advent
of digital
reservoir

may
be classified
under
one
of
four
categories:
Capillary
models
-
Are
based
on the
assumption
that a
porous medium
consists
of
a
bundle
of capillary
tubes
of
various
diameters
with a
fluid
path length
longer than
the

The
models may
be described
as
being
divided
into a
large
number
of
thin
slices
by
planes
perpendicular to the
axes
of
the tubes.
The slices
are
imagined
to
be
rearranged
and
reassembled
randomly.
Again,
statistical
models

most
successful
approximations.
Netwoik
models
-
Are
frequently
based
on the
modeling
of fluid
flow in
porous media
using a
network
of electric
resistors
as
an analog
computer.
Network
models
are
probably
the best
tools
for understanding
fluid
flow

relate
several
laboratory-
measured
parameters to
rock
permeability
was the
Kozeny-Carmen
equation.2
This equation
expresses
the
permeability
of a
porous material
as a
function
of the
product of the
effective
path length
of the
flowing
fluid and
the
mean
hydraulic
radius
of the

capillary
tubes
of
varying sizes.
Several
authorsa-r6
adapted
the
relations
developed
by Kozeny-Carmen
and Purcell
to
the
computation
of
relative
permeability.
They
all
proposed models
on
the basis
of the
assumption
that
a
porous medium
consists
of a bundle

tubes.
They tried
to
determine
tortuosity
empirically
in order
to
obtain
a close
approximation
of
experimental
data.
II. RAPOPORT
AND LEAS
Rapoport
and
Lease
presented two
equations
for
relative
permeability
to
the
wetting
phase.
16 Relative
Permeabilin

t'ot
,['*'
(tj) (T#)'
and
.['*'
P.
dS
k,*,(min)
:
(ti
-
j;
)'
fs- fS*,
I
P.ds+
|
R.as
Jr'Jr
(2)
where
S- represents
the minimum
irreducible
saturation of
the wetting
phase
from
a drainage
capillary

GATES. LIETZ. AND
FULCHER
Gates and Lietzs
developed
the following
expression based
on Purcell's
model for wetting-
phase
relative permeability:
t. _
K.*r
-
Fulcher
et al.,as have
investigated
the influence
of
capillary number
(ratio
of viscous
to
capillary
forces)
on two-phase
oil-water relative permeability
curves.
IV. FATT,
DYKSTRA,
AND BURDINE

of
the
porous
medium)
to be a function
of
saturation.
They
assumed
that
the radius
of the path
of
the
conducting pores
was
related
to the lithology
factor,
tr,
by the
equation:
ru
I$
(3)
u
hcre
r
iun
-tr.r

.,
,l
:r:nJlC
J:
.i.'1tlnS
F'-':':l(rn-
i
li-
r
:'.
:,
r\ll\
ir5
::.r
hasic
srr: :ercd a
J
i:;
Frrttus
i
f;i:.
,'l
the
Table I
CALCULATION OF
WETTING.PHASE RELATIVE
PERMEABILITY BASED
ON
THE FATT AND
DYKSTRA EQUATION

:
49.2.
n.25
7.88
5.54
3.80
2.49
t.50
0.75
0.30
0.20
100.0
70.0,
49.2b
33.8
22.1
13.3
6.1
2.7
0.4
_l
where r represents
the radius of a
pore,
a and b represent material constants,
and }, is a
function of saturation.
The equation for the wetting-phase relative
permeability,
k.*,, reported

Equation
5 to
They stated that their
equation
fit
their own data as well
as the data of
Gates and
Lietz
more
accurately than other
proposed
models.
The
procedure
for
the calculation of relative
permeability
from
capillary
pressure
data is
illustrated
by Table I and
the
results
are shown in Figures
I
and 2.
Burdine'3

and using the symbol tr*, for
the
wetting-phase
tortuosity
factor when
two
phases
are
present,
a tortuosity ratio can be defined
as
ft*'
ds
Jo
P:
TF
(6)
r-rl
T
tr.*,
:
;
(7)
r-l)
l8
Relative
Permeabilitv
of Petroleum
Reservoirs
9

t
ds/(P.)r
k.*t
:
(tr.*.)'
rl
t
ds/(p")l
In a similar
fashion,
the
relative
permeability to
the
nonwetting
phase
can
utilizing
a
nonwetting-phase
tortuosity
ratio,
tr,,*,,
then
Burdine
has
shown
that
(9)
be expressed

^
JS*t
k.n*,:
(trrn*,)'
J"
ds/(P.)2
S*,-
S-
Arwt
-
(l
t)
1-S-
l9
r60
r50
r40
r30
t20
ll
roo
90
70
60
50
40
30
20
to
o5

saturation.
The
nonwetting
phase tortuosity
can be
approximated
by
\-^ ,.:
.
Sn*t
S'
(12)
rnwt
l-s*-s"
where
S.
is the
equilibrium
saturation
to the
nonwetting
phase.
The
expression
for the
wetting
phase
(Equation
9)
fit the

relative
permeability.
Their
equations
can
be expressed
as
follows:
It
I
Pc3
|
(Cm
Hqi3
fa::
thcn
r9)
f3
.
r
lli'rred
rl0)
rll)
fs"
k,,,:
(iil'
J
os"rp;
/'
or",rl